# project-euler / project-euler / 147 / euler-147.pl

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200``` ```#!/usr/bin/perl use strict; use warnings; use 5.014; use Test::More; =head1 DESCRIPTION In a 3x2 cross-hatched grid, a total of 37 different rectangles could be situated within that grid as indicated in the sketch. There are 5 grids smaller than 3x2, vertical and horizontal dimensions being important, i.e. 1x1, 2x1, 3x1, 1x2 and 2x2. If each of them is cross-hatched, the following number of different rectangles could be situated within those smaller grids: 1x1: 1 2x1: 4 3x1: 8 1x2: 4 2x2: 18 Adding those to the 37 of the 3x2 grid, a total of 72 different rectangles could be situated within 3x2 and smaller grids. How many different rectangles could be situated within 47x43 and smaller grids? =cut # A 2-dimensional cache. # \$Rects_Coefficients_Cache[\$rect_x][\$rect_y] = { min => \$min, offset => \$offset}; # Formula for step is calculated at 2*\$board_dim + \$offset my @Rects_Coefficients_Cache; # A 4-dimensional cache: # 0 - board large dim. # 1 - board small dim. # 2 - rect large dim. # 3 - rect large dim. # Each value is: # { # # Cache for the boards: # \$Boards_Cache[\$board_x][\$board_y] = { 'num_unique' => , 'rects' => \@rects, } # rects is: # \$rects[\$rect_x][\$rect_y] my @Boards_Cache; sub get_unique_rects { # \$x >= \$y >= \$rect_x , \$rect_y my (\$x, \$y, \$rect_x, \$rect_y) = @_; my \$board_struct = (\$Boards_Cache[\$x][\$y] //= +{}); if (! defined(\$board_struct->{num_unique})) { # Calculate the unique rectangles. \$board_struct->{rects} //= []; } return \$board_struct; } sub round_two_up { my (\$n) = @_; return ((\$n&0x1) ? (\$n+1 ): \$n); } sub diag_rects { my (\$xx, \$yy, \$rect_x, \$rect_y) = @_; my \$x_even_start = \$rect_x; my \$x_even_end = ((\$xx<<1) - \$rect_y); if (\$x_even_end < 0) { return; } my \$y_even_end = ((\$yy<<1) - (\$rect_x+\$rect_y)); my \$x_even_end_norm = (\$x_even_end & (~0x1)); my \$x_even_start_norm = round_two_up(\$x_even_start); my \$even_count = 0; # For the even diagonals. if (\$x_even_end_norm >= \$x_even_start_norm and \$y_even_end >= 0) { \$even_count = ( (((\$x_even_end_norm - \$x_even_start_norm) >> 1) + 1) * ((\$y_even_end >> 1)+ 1) ); } my \$x_odd_end = ((\$x_even_end&0x1)?\$x_even_end:(\$x_even_end-1)); my \$x_odd_start = (\$x_even_start|0x1); my \$y_odd_end = (\$y_even_end - 2); my \$odd_count = 0; if (\$x_odd_end >= \$x_odd_start and \$y_odd_end >= 0) { \$odd_count = (( ( ((\$x_odd_end - \$x_odd_start) >> 1) + 1) * (((\$y_odd_end>>1)+1)) ) << (\$rect_x == \$rect_y ? 0 : 1) ); } return (\$even_count, \$odd_count); } sub get_total_rects { my (\$x, \$y) = @_; my \$sum = 0; foreach my \$xx (1 .. \$x) { foreach my \$yy (1 .. \$y) { \$sum += \$xx * \$yy * (\$x - \$xx + 1) * (\$y - \$yy + 1); } } my \$diag_sum = 0; foreach my \$xx (1 .. \$x) { foreach my \$yy (1 .. \$y) { foreach my \$rect_x (1 .. (\$xx<<1)) { RECT_Y: foreach my \$rect_y (1 .. (\$yy<<1)) { if (my (\$even_count, \$odd_count) = diag_rects(\$xx, \$yy, \$rect_x, \$rect_y)) { \$diag_sum += \$even_count + \$odd_count; } else { last RECT_Y; } } } } } return {num_straight_rects => \$sum, num_diag_rects => \$diag_sum}; } sub test_diag { local \$Test::Builder::Level = \$Test::Builder::Level + 1; my (\$xx, \$yy, \$diag_x, \$diag_y, \$want_ret_aref, \$blurb) = @_; my @have_ret = diag_rects(\$xx, \$yy, \$diag_x, \$diag_y); is_deeply( \@have_ret, \$want_ret_aref, \$blurb ); } if (1) { test_diag(1,1,1,1, [0, 0], "1,1,1,1"); test_diag(2,1,1,1, [1, 0], "1,2,1,1"); test_diag(1,2,1,1, [0, 1], "1,2,1,1"); test_diag(3,1,1,1, [2, 0], "3,1,1,1"); test_diag(1,3,1,1, [0, 2], "1,2,1,1"); test_diag(4,1,1,1, [3, 0], "3,1,1,1"); test_diag(1,4,1,1, [0, 3], "1,2,1,1"); test_diag(2,2,1,1, [2, 2], "2,2,1,1"); done_testing(); } my (\$x, \$y) = @ARGV; # my \$num_rects = get_rects(47, 43, 47, 43); my \$rects_struct = get_total_rects(\$x, \$y); my \$num_straight_rects = \$rects_struct->{num_straight_rects}; my \$num_diag_rects = \$rects_struct->{num_diag_rects}; print "Straight Rects = \$num_straight_rects\n"; print "Diag Rects = \$num_diag_rects\n"; print "Total sum = " , \$num_straight_rects + \$num_diag_rects, "\n"; ```